A possible future line of research consists in analyzing the strategic aspects of the aggregation of permutation processes. It is important to guarantee the non-manipulability of this decision making process. For example, there should not exist advantages from lying about preferences (this property is known in the literature as strategy-proofness [1]}[2]}) or from voting several times (known in the literature as anonymity-proofness [3]}). The results about avoiding manipulations are rather negative, since there exist very few rules satisfying non-manipulability, when considering unrestricted domains. We believe that in the case of the aggregation of permutation processes, new rules can be found overcoming these problems.


Ethics Declaration
Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Voting rules
Black's rule:The Black and Dodgson rules described are called Revised Black and Simplified Dodgson rules in [4]}. this rule is applied in two stages. In the first stage, the candidates that beat or tie with every other candidate are selected. If there are not such winners, each candidate receives a score depending on the positions on every ranking (the better the position the higher the score). The alternative with the largest overall score is elected.
Dodgson's rule: a Condorcet winner is a candidate that beats every other alternative. The rule computes the minimum number of times that it is necessary to swap two adjacent alternatives on some rankings in order to make each candidate a Condorcet winner. The alternative that requires the minimum steps is considered the winner.
Young's rule: this rule deletes candidates in order to make an alternative the Condorcet winner. The candidate that requires less deletions is declared the winner.
Kemeny's rule: this rule selects the most preferred alternative of the rankings that minimize the number of pairs of candidates that are ranked opposite by all the voters.
Nanson's rule: as in Black's rule, a score is given to the candidates according to their position. Every candidate with a score below the average score is deleted. The process is repeated with the candidates left, an so on. The rule selects the undeleted candidates.
Minimax rule: the candidate whose maximum losses in the paired comparisons are the least is declared winner.
Fishburn's rule: this rule checks out whether everything that beats x also beats y under simple majority, and if
x beats or ties something that beats y, then x is ‘better than’ y under simple majority. The rule selects the best alternatives under this notion.
Bucklin's rule: the Bucklin score of an alternative \(a\) , \(B(a)\) , is the minimum \(k\)  such that \(a\)  is among the first \(k\)  positions in the majority of input votes. The Bucklin rule selects the alternatives with the least Bucklin scores.
Scoring rules: these rules have an associated weights vector \( (\alpha _1, \alpha _2, \ldots , \alpha _{|A|})\) , where \(\alpha _i\ge \alpha _{i+1}\)  for \(i=1, \ldots , |A|-1\) . An alternative \(a\)  in a profile \(\sigma _j\)  gets \(\alpha _i\)  points if \(\sigma _j(a)=i\) . The score of an alternative is the sum of all the points across all voters. The candidate with the most overall points is declared the winner. The Plurality and the Antiplurality rules are examples of scoring rules with associated weights vector \((1, 0, \ldots , 0)\)  and \((1, \ldots , 1, 0)\)  respectively. We say that a scoring rule has an associated decreasing weights vector if \(\alpha _i> \alpha _{i+1}\)  for \(i=1, \ldots , |A|-1\) .

Proofs
Proposition  REF
Consider the profile \(\sigma =(a>_1 b>_1 c>_1 d>_1 e, e>_2 d>_2 a>_2 c>_2 b, b>_3 c>_3 d>_3 e>_3 a)\) . We have that \(a\rhd ^{pm}_{\pi }b\) .
Black's rule: there is no alternative that beats or tie every other alternative. Then we use a scoring rule with an associated decreasing weighted vector, for example \((5,4,3,2,1)\) . We obtain that \(score(a)=9\)  and \(score(b)=10\) , with \(b\)  getting the largest score. Thus we have that \(b\in Black(\pi )\)  and \(a\notin Black(\pi )\) .
Dodgson's rule: alternative \(b\)  needs 1 swap in the first vote to become the Condorcet winner, while all the other alternatives require 2 or more. Then \(b\in Dodgson(\pi )\)  and \(a\notin Dodgson(\pi )\) .
Young's rule: alternative \(b\)  needs alternative \(a\)  to be eliminated in order to become the Condorcet winner, while all the other alternatives need 2 or more alternatives to be eliminated. Then \(b\in Young(\pi )\)  and \(a\notin Young(\pi )\) .
Kemeny's rule: the ranking that minimizes the number of candidates that are ranked opposite by all the voters is \(b>c>d>a\) . Then \(b\in Kemeny(\pi )\)  and \(a\notin Kemeny(\pi )\) .
Consider the profile \(\sigma =(e>_1 a>_1 b>_1 c>_1 d, b>_2 c>_2 d>_2 e>_2 a)\) . We have that \(a\rhd ^{pm}_{\pi }b\) .
Nanson's rule: we use a scoring rule with an associated decreasing weighted vector, for example \((5,4,3,2,1)\) . During the first round, \(a\)  and \(d\)  are eliminated. In the second round \(c\)  is eliminated. Finally we have that \(b\)  and \(e\)  are selected. Then \(b\in Nanson(\pi )\)  and \(a\notin Nanson(\pi )\) .
Consider the profile \(\sigma =(d>_1 c>_1 a>_1 b, b>_2 c>_2 d>_2 a, d>_3 c>_3 a>_3 b, a>_4 b>_4 c>_4 d, b>_5 c>_5 d>_5a)\) . We have that \(a\rhd ^{pm}_{\pi }b\) .
Minimax rule: alternative \(b\) , \(c\)  and \(d\)  have 2 as a maximum loss while \(a\)  has 3. Then \(b\in Minimax(\pi )\)  and \(a\notin Minimax(\pi )\) .
Consider the profile \(\sigma =(a>_1 b>_1 c>_1 d, b>_2 c>_2 d>_2 a)\) . We have that \(a\rhd ^{pm}_{\pi }b\) .
Fishburn's rule: in this profile \(b\)  beats \(c\)  and \(d\) , ties with \(a\)  and beat all the alternative that beat or tie with \(a\) . Then \(b\in Fishburn(\pi )\)  and \(a\notin Fishburn(\pi )\) .

Proposition  REF
Let \(a\rhd ^{pd}_{\pi }b \)  and \(b\in Bucklin(\pi )\) . Let \(B(b)=t\) , that is, \(s_{t}(b)=\alpha \ge [\frac{n+1}{2}]\)  Because of position dominance, we have that \(s_{t}(a)\ge s_{t}(b)=\alpha \) . Then, \(B(a)\le t\) . So \(a\in Bucklin(\pi )\) .
The following example shows that it is not strong-pd-eff. Let \(\pi \)  be the preference profile such that \(\sigma =(a>_1 b>_1 c,a>_2 b>_2 c,b>_3 a>_3 c,c>_4 a>_4 b)\) . Then \(B(a)=B(b)=2\) , \(B(c)=3\) , \(s_{1}(a)=2\) , \(s_{1}(b)=\) , \(s_{2}(a)=4\)  and \(s_{2}(b)=3\) . So, we have that \(a\rhd ^{pd}_{\pi }b\)  and \(b\ntriangleright ^{pd}_{\pi }a\)  but \(Bucklin(\pi )=\lbrace a,b\rbrace \) .

Proposition  REF
For ease of demonstration we assume, without loss of generality, that \(A=\lbrace a,b,c\rbrace \) . The score of alternative \(a\)  is given by \(Score(a)=(s_{1}(a),s_{2}(a)-s_{1}(b),3-s_{2}(a))\cdot (\alpha _{1},\alpha _{2},\alpha _{3})=(\alpha _{1}-\alpha _{2})s_{1}(a)+(\alpha _{2}-\alpha _{3})s_{2}(a)+3\alpha _{3}\)  with \(\alpha _{i}-\alpha _{i+1}\ge 0\)  for \(i=1,2\) . From the fact that \(a\rhd ^{pd}_{\pi }b \)  we have that \(s_{j}(a)\ge s_{j}(b)\)  for \(j=1,2\) . Then \(Score(a)\ge Score(b)\) . If \(b\)  is such that \(b\in f(\pi )\) , then \(a\in f(\pi )\)  as it has more or equal points than \(b\) .
Proposition  REF
We assume again that \(A=\lbrace a,b,c\rbrace \) . And consider the same argument as in the proof of Proposition REF . The main difference is that \(\alpha _{i}-\alpha _{i+1}>0\)  for \(i=1,2\) . Then if \(a\rhd ^{pd}_{\pi }b \)  and \(b\ntriangleright _{\pi }^{pd}a\) , we have that \(Score(a)> Score(b)\)  and then \(b\notin f(\pi )\) . If instead \(b\rhd ^{pd}_{\pi }a\) , then \(Score(a)= Score(b)\)  and both alternatives are selected if they have the most overall points.
